| L(s) = 1 | − 1.37e11·4-s − 5.88e15·7-s − 5.96e19·13-s + 1.88e22·16-s − 5.87e22·19-s − 7.27e25·25-s + 8.08e26·28-s − 6.66e27·31-s − 1.67e29·37-s − 2.50e30·43-s + 1.60e31·49-s + 8.20e30·52-s − 1.48e33·61-s − 2.59e33·64-s − 1.06e34·67-s − 5.41e33·73-s + 8.07e33·76-s − 5.10e33·79-s + 3.51e35·91-s + 1.02e37·97-s + 9.99e36·100-s − 1.91e37·103-s − 3.90e37·109-s − 1.11e38·112-s + ⋯ |
| L(s) = 1 | − 4-s − 1.36·7-s − 0.147·13-s + 16-s − 0.129·19-s − 25-s + 1.36·28-s − 1.71·31-s − 1.62·37-s − 1.51·43-s + 0.864·49-s + 0.147·52-s − 1.39·61-s − 64-s − 1.76·67-s − 0.182·73-s + 0.129·76-s − 0.0399·79-s + 0.201·91-s + 1.79·97-s + 100-s − 1.10·103-s − 0.793·109-s − 1.36·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+37/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(19)\) |
\(\approx\) |
\(0.1114726454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1114726454\) |
| \(L(\frac{39}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + p^{37} T^{2} \) |
| 5 | \( 1 + p^{37} T^{2} \) |
| 7 | \( 1 + 5882725491086809 T + p^{37} T^{2} \) |
| 11 | \( 1 + p^{37} T^{2} \) |
| 13 | \( 1 + 59698451722484942635 T + p^{37} T^{2} \) |
| 17 | \( 1 + p^{37} T^{2} \) |
| 19 | \( 1 + \)\(58\!\cdots\!07\)\( T + p^{37} T^{2} \) |
| 23 | \( 1 + p^{37} T^{2} \) |
| 29 | \( 1 + p^{37} T^{2} \) |
| 31 | \( 1 + \)\(66\!\cdots\!84\)\( T + p^{37} T^{2} \) |
| 37 | \( 1 + \)\(16\!\cdots\!61\)\( T + p^{37} T^{2} \) |
| 41 | \( 1 + p^{37} T^{2} \) |
| 43 | \( 1 + \)\(25\!\cdots\!68\)\( T + p^{37} T^{2} \) |
| 47 | \( 1 + p^{37} T^{2} \) |
| 53 | \( 1 + p^{37} T^{2} \) |
| 59 | \( 1 + p^{37} T^{2} \) |
| 61 | \( 1 + \)\(14\!\cdots\!01\)\( T + p^{37} T^{2} \) |
| 67 | \( 1 + \)\(10\!\cdots\!35\)\( T + p^{37} T^{2} \) |
| 71 | \( 1 + p^{37} T^{2} \) |
| 73 | \( 1 + \)\(54\!\cdots\!63\)\( T + p^{37} T^{2} \) |
| 79 | \( 1 + \)\(51\!\cdots\!23\)\( T + p^{37} T^{2} \) |
| 83 | \( 1 + p^{37} T^{2} \) |
| 89 | \( 1 + p^{37} T^{2} \) |
| 97 | \( 1 - \)\(10\!\cdots\!49\)\( T + p^{37} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.44195071296549842250580151705, −9.568926420396444085524282826649, −8.762822406578952685699272755079, −7.42293027033612293995121984043, −6.19501625171201676209516751016, −5.16600432144325685699575902603, −3.88899081179603352631935101173, −3.16425451817598054258380062381, −1.64267397985451082069483491154, −0.13470806420437748892904581416,
0.13470806420437748892904581416, 1.64267397985451082069483491154, 3.16425451817598054258380062381, 3.88899081179603352631935101173, 5.16600432144325685699575902603, 6.19501625171201676209516751016, 7.42293027033612293995121984043, 8.762822406578952685699272755079, 9.568926420396444085524282826649, 10.44195071296549842250580151705
